Forward-Backward Doubly Stochastic Differential Equations

Abstract

Stochastic calculus is a branch of mathematics that operates on stochastic pro- cesses. It allows a consistent theory of integration to be dened for integrals of stochastic processes with respect to semi-martingales. It is used to model systems that behave randomly. These include stochastic dierential equation (SDEs). In 1944 Kiyosi It^o, a Japanese mathematician (1915-2008), introduced the stochastic integral and a formula, known since then as It^o's formula. The best-known stochas- tic process to which stochastic calculus is applied is the Brownian motion/Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion, as described by Louis Bachelier in 1900 and by Albert Einstein in 1905, and other physical diusion processes in space of particles subject to random forces. Since 1970s, Brownian motion has been widely applied in nancial mathematics and economics to model the evolution in time of stock prices, options, and bond interest rates.